The Stacks project

Lemma 73.9.5. Let $S$ be a scheme, and let $X$ be an algebraic space over $S$. Suppose that $\mathcal{U} = \{ f_ i : X_ i \to X\} _{i \in I}$ is an fpqc covering of $X$. Then there exists a refinement $\mathcal{V} = \{ g_ i : T_ i \to X\} $ of $\mathcal{U}$ which is an fpqc covering such that each $T_ i$ is a scheme.

Proof. Omitted. Hint: For each $i$ choose a scheme $T_ i$ and a surjective étale morphism $T_ i \to X_ i$. Then check that $\{ T_ i \to X\} $ is an fpqc covering. $\square$

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